Ha‘i 'break, snap' - Honolulu Star-Bulletin music critic John Berger explains it in an April 2002 story headlined That LeAnn Rimes yodel translates to Hawaiian ha‘i.
First, about ha'i. The relevant translation relates to a style in which singers voice a break when moving between their lower register ("chest voice") and upper register ("head voice"). Hawaiian falsetto singers use this technique to emphasize or add emotional intensity to a phrase or passage, whereas traditional European-American falsetto singers try to eliminate any hint of it.Puana 'refrain, theme, or keynote of a song' - Puana is the key word in the phrase, Ha‘ina ‘ia mai ana ka puana. Pukui & Elbert's Hawaiian Dictionary (1986) defines puana thus:
"When I was studying musical theory (in college), my voice teacher and I spent five years trying to smooth that break out," [Amy Hanaiali'i] Gilliom said.
Certain women, like LeAnn Rimes, sing ha'i, she said, singing a few bars of Rimes' hit "Blue" to prove her point.
Keawe agreed. "I've heard that recording and I hear her singing, and she sings like us with the ha'i. You can sing ha'i in any language."
Attack or beginning of a song; in music, the tonic or keynote; to begin a song; summary refrain, as of a song, usually at or near the beginning of a song; theme of a song.The whole phrase is rather awkwardly translated as
tell the summary refrain (this line followed by the refrain is at the end of many songs or precedes the name of the person in whose honor the song was composed).